We develop an inverse geometric optimization technique that allows the derivation of optimal and robust exact solutions of low-dimension quantum control problems driven by external fields. We determine in the dynamical variable space optimal trajectories constrained to robust solutions by Euler-Lagrange optimization; the control fields are then derived from the obtained robust geodesics and the inverted dynamical equations. We apply this method, referred to as robust inverse optimization (RIO), to design optimal control fields producing a complete or half population transfer and a not quantum gate robust with respect to the pulse inhomogeneities. The method is versatile and can be applied to numerous quantum control problems, e.g., other gates, other types of imperfections, Raman processes, or dynamical decoupling of undesirable effects.